Chapter 10, Problem 22RP

a)

To determine

To show : WXYZ is a rhombus.

Answer to Problem 22RP

WXYZ is a parallelogram. WZ and YZ are tangent segment.

  

Explanation of Solution

Given Information :

The length of $2$ tangents drawn from a point outside the circle are equal.

Formula Used:

WZ = YZ (both are tangents)

Since WXYZ is a parallelogram and adjacent sides are equal thus WXYZ is a rhombus.

b)

To determine

To find : The value of $m\angle Z$

Answer to Problem 22RP

  $m\angle Z=60°$

Explanation of Solution

Given Information :

WXYZ is a parallelogram. WZ and YZ are tangent segments.

Formula Used:

The length of $2$ tangents drawn from a point outside the circle are equal.

Calculation :

In $\Delta XPY$ and $\Delta ZPY$

XP = ZP

[Diagonals of rhombus]

YP is common

XY = ZY

[Sides of rhombus]

  $\begin{array}{l}\therefore \Delta XPY\cong \Delta ZPY\\ \therefore \angle PXY=\angle PZY=a\end{array}$

Now OX = OY = radius

  $\therefore \Delta OXY$ is isosceles.

  $\therefore \angle OXY=\angle OYX=a$

Diagonals of a rhombus bisect each other.

  $\therefore \angle YZW=2\angle YZP$

c)

To determine

To find : Perimeter of WXYZ.

Answer to Problem 22RP

Perimeter = $60$

Explanation of Solution

Given Information :

The given information is WY = $15$ .

Formula Used:

Perimeter of WXYZ = Sum of all side.

Calculation :

As WXYZ is a rhombus.

Perimeter of WXYZ = WX + XY + YZ + W

  $\begin{array}{l}=15+15+15+15\\ =60\end{array}$

d)

To determine

To find : The value of XZ

Answer to Problem 22RP

XZ = $25.98$

Explanation of Solution

Given Information :

The given information is WY = $15$

Formula Used:

The point of intersection of the diagonals of a bisect each diagonal.

Calculation :

  $\begin{array}{l}XZ=2PZ\\ PZ=YZ\cos 30°\\ =15\times 0.86=12.99\\ XZ=2\times 12.99=25.98\end{array}$